Theorem syl3anr1 1443

Description: A syllogism inference. (Contributed by NM, 31-Jul-2007.)

Hypotheses

Ref	Expression
syl3anr1.1	⊢ (𝜑 → 𝜓)
syl3anr1.2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

Assertion

Ref	Expression
syl3anr1	⊢ ((𝜒 ∧ (𝜑 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

Proof of Theorem syl3anr1

Step	Hyp	Ref	Expression
1		syl3anr1.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	3anim1i 1170	. 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜏))
3		syl3anr1.2	. 2 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
4	2, 3	sylan2 605	1 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  prmidlc2  21526  btwnconn1lem4  36621  pridlc2  38783  atmod1i1  40691  prmdvdsfmtnof1lem2  48397